Transcript Operations with Vectors

More On Vectors
By Mr. Wilson
September 21, 2012
Honors Geometry, FWJH
What vector
describes me
walking from
my Car (C) to
Lunch (L) ?
From Lunch (L)
to my School
class (S)?
From School (S)
back to my
car (C)?
CL   12,5
LS  5,9
SC  7,14
• So I walked along all these vectors and
yet I wound up right where I started?
• What if I walked these same vectors in a
different order? Will I still get back to
my car?
Adding Vectors
• In general, when adding vectors <a1, b1> and
<a2, b2>, they result in the new vector
<a1 + a2, b1 + b2>
• It doesn’t matter what order you add the
vectors. You end up at the same spot.
We’ve Been Doing This Already!
• What do we mean by < 3 , -4 >?
Go right 3 in the x-direction
Go down 4 in the y-direction
< 3 , -4 > = < 3 , 0 > + < 0 , -4 >
SLATES TIME!
Add the following vectors:
< 12, -3 > + < 2, 0 > = ?
< -5, 2 > + < -4, -3 > = ?
<10,0> + <0,-10> + <-10,0> + <0,10>
=?
Multiplying a Vector by a Number
• We can stretch, squish, or flip a vector around
by multiplying it by a scalar (factor)


If u  a, b , then Nu  Na, Nb
• Example:
3< -2 , 4 > = < 3(-2) , 3(4) > = < -6 , 12 >
Notes on Scalar Multiplying
• If |N| > 1, then the vector is getting stretched
out. Its length is increasing.
• If |N| < 1, then the vector is getting squished
in. Its length is decreasing.
• If N < 0, then the vector is now going in the
opposite direction
Multiply a Vector by… Another Vector?

• The Dot Product of two vectors u  a1 ,b1
and v  a ,b is given by
2
2
 
u  v  a1a2  b1b2
Note that the dot product of two vectors is a
SCALAR (NUMBER), NOT A VECTOR
Notes on Dot Product
• If the two vectors are parallel, we have
 


| u  v | (length of u )(length of v )
• If the two vectors are perpendicular,
 
u v  0
• This comes up in Trigonometry, Physics,
Multi-Dimensional Calculus
SLATES AGAIN!
Are these vectors parallel, perpendicular, or
neither?
< 6 , -8 > and < -3, 4 > ?
< 2, -5 > and < 5, -2 >?
< 0 , 3 > and < -9 , 0 > ?
Find a vector perpendicular to <9,7>.